
where the expectation is over the joint density \(p_{\mathbf {\theta }}(\mathbf {x}^{\prime },y^{\prime })\) , sampling from which is challenging due to the intractable \(Z(\mathbf {\theta })\) , and \(\beta \)  is a hyperparameter that balances the contributions from the two terms. Different value of \(\beta \)  has a significant impact to the performance. From our experiments, we find that \(\beta \!=\!0.5\)  works very well in all our cases. Therefore, we set \(\beta \!=\!0.5\)  as the default value. Notably, the two terms of our GMMC (Eq. REF ) are both defined on the same energy function \(E_{\mathbf {\theta }}(\mathbf {x},y)\) , while the two terms of JEM [1]} are defined on \(p(\mathbf {x})\)  and \(p(y|\mathbf {x})\) , respectively, which are computationally more expensive and might incur training instability as we will discuss in Sec. REF .
